Time averaging of weak values using the quantum transition path time probability distribution enables us to establish a general uncertainty principle for the weak values of two not necessarily Hermitian operators. This new principle is a weak value analog of the Heisenberg-Robertson strong value uncertainty principle. It leads to the conclusion that it is possible to determine with high accuracy the simultaneous mean weak values of non-commuting operators by judicious choice of the pre- and post-selected states. Generally, when the time fluctuations of the two weak values are proportional to each other there is no uncertainty limitation on their variances and, in principle, their means can be determined with arbitrary precision even though their corresponding operators do not commute. To exemplify these properties we consider specific weak value uncertainty relations for the time-energy, coordinate-momentum and coordinate-kinetic energy pairs. In addition we analyze spin operators and the Stern-Gerlach experiment in weak and strong inhomogeneous magnetic fields. This classic case leads to anomalous spin values when the field is weak. However, anomalous spin values are also associated with large variances implying that their measurement demands increased signal averaging. These examples establish the importance of considering the time dependence of weak values in time of flight scattering experiments.
Weak values, 'negative probability' and the uncertainty principle
量子跃迁可以看作是各种路径（例如费曼路径）之间干涉的结果，这些路径可以用变量$f$标记。试图确定f的值而不破坏通路之间的相干性会产生$\bar{f}$的弱值。我们表明$\bar{f}$是一个平均值，得到的振幅分布，可以在一般情况下，采取负值，按照不确定原理，不需要包含信息的实际范围内的值$f$，有助于过渡。我们还证明了这种交替分布的矩具有一些不寻常的性质，这可能会导致对弱测量结果的误解。我们提供了一个详细的分析，弱测量后选择和不后选择。例子包括双缝衍射实验、弱冯·诺依曼和类冯·诺依曼测量、弹性碰撞的穿越时间、相位时间、局部角动量（LAM）和{ \it Aharonov et al}的“三箱情况”
Tight qubit uncertainty relations studied through weak values in neutron interferometry
Atominstitut, TU Wien（维也纳工业大学原子研究所）；Department of Applied Physics, Hokkaido University（北海道大学应用物理系）；Graduate School of Advanced Science and Engineering, Hiroshima University（广岛大学先进理工学研究科）；Institut Laue Langevin（劳厄-朗之万研究所）
我们将Wootters和Fields最初给出的用于状态确定（断层扫描）的测量的\emph{optimality}的概念应用于\emph{pure states}的\emph{weak value tomography}。他们将测量定义为最优的，如果它们“最小化”统计误差的影响。由于技术原因，他们实际上最大化了状态平均信息，精确地量化为“错误量”的负对数。在本文中，我们优化的状态平均信息以及错误量。我们证明了，对于任意（有限）维的Hilbert空间，当后选择的基地是\emph{mutually unbiased}关于被测量的可观察的特征向量的弱值测量的品种是最佳的。我们证明了一些重要的结果表示时，通过弱值作为坐标的状态空间的几何。我们导出了N维情形下的Kaëhler势的表达式，并利用它给出了任意自旋情形的精确处理。
